Fixed Inc

Front 1

Spot rates are yields on what type of bond

Back 1

. zero coupon bonds

Front 2

A fwd rate is an interest rate agreed today for a loan to be made in future

Back 2

.

Front 3

Price of $1 zero coupon is discount factor(Pt). Since zero coupon it iss spot is ytm(St).

Pt=

Back 3

1 / (1+St)^maturity

Front 4

Term structure of fwd rates is called fwd curve

Back 4

.

Front 5

Fwd rate of a k year loan starting in j years

Back 5

K+j

Or

1/ (1+ (j+k)^k)

Front 6

Find ytm using pv function on calc with payments per year and face value with final coupon at end then use that value for PV with N and PMt and FV at 1k and compute for IY

Back 6

.

Front 7

Expected retun/ex ante will be equal to bond yield only when

Back 7

1 held to maturity

2. All payment made in full on time

3. All payments reinvested at original ytm

Front 8

Realized returns on bonds refers to actual return based on rates that reinvestment are made

Back 8

.

Front 9

What is fwd price 2 yrs from now for $1 par zero coupon 3 yr bond with

2yr spot of 4%

5yr spot of 6%

Back 9

1/ 1+.04 ^2=.9246

1/ 1+.06 ^5 =.7473

.7473/.9246=.8082

.8082 is price agreed today to pay in 2yrs for a 3yr bond that will pay $1 at maturity

Front 10

2 and 5 yr spot rates are 4% and 6%

What is 3yr fwd rate for a loan starting 2 yrs from now

Back 10

(1+.06)^5 / (1+.04)^2 = 1.23

Answer 7.35%??

Front 11

Bootstrapping

Compute spot par curve

1yr 1%

2yr 1.25%

3yr 1.5%

Back 11

S1=1%

101.25/(1.0125^2)=98.77

101.25/98.77=1.0252

SqRt 1.0252 -1= S2= 1.252%

For 3 yr

1.5/1.01=1.485

1.5/ 1.0252^2=1.46

=2.9483(divs added together and subtract from 100 whichbis 97.052 or just discount 100+divy? Which is

101.5/ 1.015^3= 97.067

101.5/97.052=1.0458

1.0458^(1/3)=S3=1.0151-1=1.51%

Front 12

What is holding return1yr zero coupon with maturity 1 yr and spot of 3

Back 12

Price=1/1.03=.9709

It pays 1 at endnof year so

1/.9709 -1=3%

Front 13

2 yr holding return of zero coupon with maturiy of 2 with spot of 4% and EXpected spot of 5.01% in yr1

Back 13

Price=1/ 1.04^2=.9246

Price after 1 yr= 1/1.0501=.9523

Return=.9523/.9246=3%

Front 14

3 yr holding return of zero coupon with maturiy of 3 with spot of 5% and EXpected spot of 6.01% in yr1

Back 14

Price=1/ 1.05^3=.8638Price after 1 yr= 1/1.0601^2=.8898Return=.8898/.8638=3%

Front 15

Riding yield curve

Back 15

Buy bonds with longer maturity than his investment horizon. In an upward yield curve short term have lower rates. As bond gets closer to maturity(rolls down yield curve) it is valued with lower yields and higher prices

Front 16

In 07-08 Borrow at short term rates to buys LT bonds with higher yields. But risk is that spot rate could increase

Back 16

.

Front 17

Maturity - 5 yrs

Yield - 3

Price -100

Maturity - 25 yrsYield - 5Price -72

Maturity - 30 yrsYield - 5.5Price -64

Could buy 5yr and make 3% coupon.

Or buy 30yr for 64, sell in 5 yrs for 72 and make more than coupon

Back 17

.

Front 18

Plain vanilla swap

Back 18

1 party makes payments on fixed

1 party makes payments on floating

Front 19

Prefer swap rate curve to gov bond yield curve for:

Back 19

Swaps reflect credit risk of commercial banks not govs

Swap market not regulated which makes differing countries swaps more comparable

Swap curve has more maturities

Front 20

Wholesale banks more likely use swap rates

Retail use gov bond yield

Back 20

.

Front 21

Given libor spot rate curve compute fixed rate for year 2

Maturity 1 spot 3%

Maturiy 2. Spot 4%

Numbers arent right

A diff way is in derivitatives section

Back 21

SFR2/ 1+S1 + SFR2/(1+S2)^2 + 1/(1+S2)^2=1

SFR2/ 1.03 + SFR2/(1+04)^2 + 1/(1+.04)^2=1

SFR2/ 1.03 + SFR2/(1+04)^2 + .9246=1

SFR2/ 1.03 + SFR2/(1.0816)= .0754

SFR2 + SFR2/(1.0816)= .0754*1.03

SFR2 + SFR2/(1.0816)= .0778*1.0816

SFR2 + SFR2= .08405

.08405/2=Swap fixed rate for 2nd year

Front 22

Swap fixed rate for year 2

Back 22

(SFR2/ 1+spot1) + (1SFR2/ 1+spot2^2)+ (1/ 1+spot2^2)= 1

Front 23

Swap spread

Back 23

Amout swap rate exceeds gov yield

Front 24

Libor swap of 2.02. Us treasury 1.61. What is swap spread

Back 24

41 bps

Front 25

I spread

Back 25

Amount yield on risky bond exceeds swap rate for same maturity

Front 26

I spread calc 33h

Back 26

.

Front 27

Zspread

Back 27

Spread when added to each spot rate makes PV of bonds CF equal to MArket price

Front 28

3 yr 5% bond trades at zspread ofn100bps

Ql1yr. 1yr fwd inb1yr. 1yr fwd in 2yrs are

3%, 5.051%, 7.198%

Find price of bond

Back 28

First find spot rates

S2= 1.03*1.0501=4.02%

S3= 1.03* 1.0501* 1.07198=5.07%

Divide above by 2 and 3?

Then

5/ (1.03+.01.) + 5/ (1.0402+.1^2) + 105/ (1.0507+.01^3)= 97.33

Front 29

Ted spread

Back 29

Spread between libor and short term US

Libor reflects risk of lenigng to commercial banks. Tbills are risk free. Rising ted means investors thinks banks more likely to default

Front 30

OIS

Back 30

Overnight index swap refelcts fed funds rate

Front 31

Libor OIS spread

Back 31

Amt libor excees OIS. Ois has only minimal risk. Low spread mean high liquidity while high means banks less willing to lend

Front 32

Unbiased expectations theory or pure expectations theory

Back 32

Fwd rates exclusively are ex0ected spot rates

Investors expectations make yield curve. Every maturity must have same expected return. So if 1yr and 2 yr spot rates are 5% and 7%. Cthe 1yr fwd rate in 1 yr must be 9%. Because investing for 2yrs at 7% yields the same as 1yr at 5% and 1yr at 9%.

Front 33

Local expectations theory

Back 33

Similar to imunbiased theory but only for short term. So long term a risk premium should exist. But all bonds in short term earn risk free. This theory doesnt hold cause long bonds have higher short holding period returns.** short term HPR for long bonds outperform short term Hpr of short term bonds

Front 34

segmented markets theory

Back 34

the yield at each maturity is determined independently of the yields at other maturities; we can think of each maturity to be essentially unrelated to other maturities.

Front 35

preferred habitat theory

Back 35

imbalance between the supply and demand for funds in a given maturity range will induce lenders and borrowers to shift from their preferred habitats (maturity range) to one that has the opposite imbalance. However, to entice investors to do so, the investors must be offered an incentive to compensate.

Unlike the liquidity preference theory, under the preferred habitat theory a 10-year bond might have a higher or lower risk premium than the 25-year bond. It also means that the preferred habitat theory can be used to explain almost any yield curve shape.

Front 36

Equilibrium term structure models say changes in the term structure through the use of fundamental economic variables that drive interest rates. the two famous models discussed in the curriculum, the Cox-Ingersoll-Ross (CIR) model and the Vasicek Model, are both single-factor models. The single factor in the CIR and Vasicek model is the short-term interest rate.

Back 36

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Front 37

Cox-Ingersoll-Ross model is based on the idea that interest rate movements are driven by individuals choosing between consumption today versus investing and consuming at a later time. =

Back 37

dr=a(b−r)dt+σ√rdz

where:dr= change in the short-term interest rate

a= speed of mean reversion parameter (a high a means fast mean reversion)

b= long-run value of the short-term interest rate

r= the short-term interest rate

t= time

dt= a small increase in time

σ= volatility

dz= a small random walk movement

Front 38

Under the CIR model, volatility increases with the interest rate, as can be seen in the σ√rr dz term. In other words, at high interest rates, the amount of period-over-period fluctuation in rates is also high.

Back 38

...

Front 39

Vasicek model suggests that interest rates are mean reverting to some long-run value.

=

Back 39

dr=a(b−r)dt+σdz

Front 40

volatility in this model does not increase as the level of interest rates increase. Disadavantage is it does not force interest rates to be non nwgative

Back 40

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Front 41

Arbitrage-free models of the term structure of interest rates begin with the assumption that bonds trading in the market are correctly priced, and the model is calibrated to value such bonds consistent with their market price

Back 41

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Front 42

Ho-Lee model..assumes no arbitrage

Back 42

Ho-Lee model is calibrated by using market prices to find the time-dependant drift term θt that generates the current term structure. The Ho-Lee model can then be used to price zero-coupon bonds and to determine the spot curve. The model produces a symmetrical (normal) distribution of future rates.

Front 43

Yield curve risk refers to risk to the value of a bond portfolio due to unexpected changes in the yield curve.To counter yield curve risk, we first identify our portfolio’s sensitivity to yield curve changes using one or more measures. Yield curve sensitivity can be generally measured by effective duration, or more precisely using key rate duration, or a three-factor model that decomposes changes in the yield curve into changes in level, steepness, and curvature

Back 43

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Front 44

Effective duration

Back 44

Effective duration measures price sensitivity to small parallel shifts in the yield curve. It is important to note that effective duration is not an accurate measure of interest rate sensitivity to non-parallel shifts in the yield curve like those described by shaping risk. Shaping risk refers to changes in portfolio value due to changes in the shape of the benchmark yield curve.

Front 45

. It is important to note that effective duration is not an accurate measure of interest rate sensitivity to non-parallel shifts in the yield curve like those described by shaping risk. Shaping risk refers to :

Back 45

changes in portfolio value due to changes in the shape of the benchmark yield curve.

Front 46

Key rate duration is the sensitivity of the value of a security to changes in a single par rate, holding all other spot rates constant.

Back 46

.

Front 47

the 1-year, 5-year, and 25-year maturities, with key rate durations represented by D1 = 0.7, D5 = 3.5, and D25 = 9.5, respectively.The model for yield curve risk using these key rate durations would be:

Back 47

-.7*change in rate(or risk)--3.5*change in rate--9.5*change in rate

Front 48

An alternative to decomposing yield curve risk into sensitivity to changes at various maturities (key rate duration) is to decompose the risk into sensitivity to the following three categories of yield curve movements:Level (ΔxL) − A parallel increase or decrease of interest rates.Steepness (ΔxS) − Long-term interest rates increase while short-term rates decrease.Curvature (ΔxC) − Increasing curvature means short- and long-term interest rates increase while intermediate rates do not change.

Back 48

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Front 49

Change in value of portfolio

where DL, DS, and DC are respectively the portfolio’s sensitivities to changes in the yield curve’s level, steepness, and curvature.

Back 49

Dl*change in yield - Ds*change in yield- Dc*change in yield...9f answer was .05 it predicts an +.05% change in portfolio value.

Front 50

term structure of interest rate volatility is the graph of yield volatility versus maturity.

Volatility at the long-maturity end is thought to be associated with uncertainty regarding the real economy and inflation, while volatility at the short-maturity end reflects risks regarding monetary policy.

Back 50

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Front 51

If an investor thinks future spot rates will be below fwd rates they will buy

Back 51

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Front 52

When yield curv is upward sloping mgr will ride yield curve by holding LT rather than ST bonds. He earns in excess as the bond rolls down the yield curve

Back 52

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Front 53

For default free bond. Swap spreads have indicators of 1 for the bonds liqduidity or 2 possible misprice

Back 53

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Front 54

Libor OIS spread is indicator of well being of banking system

Back 54

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